An analogue of the Connes-Kasparov homomorphism for \(p\)-adic simple algebraic groups of split rank one (Q1907290)

Let \(G\) be a unimodular separable locally compact group acting continuously and properly on a tree \(X\), \(B\) be a stabilizer group of some edge of \(X\), \((\pi,V)\) be a representation of \(G\) such that \(\dim V^B<\infty\), where \(V^B=\{v\in V\mid\pi(h)v=v\) for all \(h\in B\}\). The aim of this paper is to construct for each of these representations an element \([J_\pi]\in K_0(C^*(G))\) which is analogous to Kasparov's one in the case of a real simple Lie group. As an application of this result an analogue of the Connes-Kasparov homomorphism for \(p\)-adic simple algebraic groups of split rank one is obtained.